Slides by Kai Arras

Last update: June 2010 1

Wolfram Burgard, Cyrill Stachniss, Maren

Bennewitz, Giorgio Grisetti, Kai Arras

Error Propagation

Introduction to Mobile Robotics

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•  Probabilistic robotics is • Representation • Propagation • Reduction of uncertainty

•  First-order error propagation is fundamental for: Kalman filter (KF), landmark extraction, KF-based localization and SLAM

Error Propagation: Motivation

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First-Order Error Propagation

Approximating f(X) by a first-order Taylor series expansion about the point X = µX

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First-Order Error Propagation

X,Y assumed to be Gaussian

Y = f(X)

Taylor series expansion

Wanted: , (Solution on blackboard)

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Y = f(X1, X2, ..., Xn)

Taylor series expansion

Wanted: , (Solution on blackboard)

First-Order Error Propagation

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First-Order Error Propagation

Y = f(X1, X2, ..., Xn)

Z = g(X1, X2, ..., Xn)

Wanted:

(Exercise)

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First-Order Error Propagation

Putting things together...

with

“Is there a compact form?...”

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Jacobian Matrix

•  It’s a non-square matrix in general

•  Suppose you have a vector-valued function

•  Let the gradient operator be the vector of (first-order) partial derivatives

•  Then, the Jacobian matrix is defined as

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•  It’s the orientation of the tangent plane to the vector-valued function at a given point

•  Generalizes the gradient of a scalar valued function

•  Heavily used for first-order error propagation...

Jacobian Matrix

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First-Order Error Propagation

Putting things together...

with

“Is there a compact form?...”

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First-Order Error Propagation

...Yes! Given

•  Input covariance matrix CX

•  Jacobian matrix FX

the Error Propagation Law

computes the output covariance matrix CY

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First-Order Error Propagation

Alternative Derivation:

Wanted: Parameter Covariance Matrix

Simplified sensor model: all , independence

Result: Gaussians in the model space

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Example: Line Extraction

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•  Second-Order Error Propagation

Rarely used (complex expressions)

•  Monte-Carlo

Non-parametric representation of uncertainties

1. Sampling from p(X)

2. Propagation of samples

3. Histogramming

4. Normalization

Other Error Prop. Techniques